Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces

Description

This book provides a detailed study of nonlinear partial differential equations satisfying certain nonstandard growth conditions which simultaneously extend polynomial, inhomogeneous and fully anisotropic growth. The common property of the many different kinds of equations considered is that the growth conditions of the highest order operators lead to a formulation of the equations in Musielak–Orlicz spaces. This high level of generality, understood as full anisotropy and inhomogeneity, requires new proof concepts and a generalization of the formalism, calling for an extended functional analytic framework. This theory is established in the first part of the book, which serves as an introduction to the subject, but is also an important ingredient of the whole story. The second part uses these theoretical tools for various types of PDEs, including abstract and parabolic equations but also PDEs arising from fluid and solid mechanics. For connoisseurs, there is a short chapter on homogenization of elliptic PDEs. The book will be of interest to researchers working in PDEs and in functional analysis. Preface Acknowledgements Contents Part I Overture Chapter 1 Introduction Chapter 2 N-Functions 2.1 Elementary Facts 2.1.1 Properties of convex functions 2.1.2 Carathéodory functions 2.1.3 The conjugate function 2.1.4 The second conjugate function 2.2 Definition of an N-Function 2.3 Refined Properties of N-Functions 2.3.1 Examples of N-functions 2.3.2 Conjugation and degeneracy 2.3.3 Remarks on isotropic functions 2.3.4 Consequences of the Δ_2-condition Chapter 3 Musielak–Orlicz Spaces 3.1 Definitions and Fundamental Properties 3.2 Embeddings L_M1 ⊂ L_M2 and L_M1 ⊂ E_M2 3.3 Function Spaces in View of the Δ_2-Condition 3.4 Topologies 3.4.1 The modular topology and uniform integrability 3.4.2 Modular density of simple functions and separability of E_M∗ 3.5 Duality (E_M)^∗ = L_M∗ 3.6 Function Spaces in PDEs 3.7 Density and Approximation 3.7.1 Condition I (general growth) 3.7.2 Condition II (at least power-type growth) 3.7.3 Between isotropic and anisotropic conditions 3.7.4 Density results 3.8 Operators and Related Musielak–Orlicz Spaces 3.8.1 Special instances 3.8.2 The meaning of the growth and coercivity conditions Part II PDEs Chapter 4 Weak Solutions 4.1 Elliptic Equations 4.1.1 Assumptions on the operator 4.1.2 The monotonicity trick in the elliptic case 4.1.3 Elliptic problems in cases M ∈ Δ_2 or M∗ ∈ Δ_2 4.1.4 Elliptic problems via the modular density approach 4.2 Parabolic equation 4.2.1 Assumptions on the operator 4.2.2 Approximation in space 4.2.3 Integration by parts formula 4.2.4 The monotonicity trick in the parabolic case 4.2.5 Bounded-data parabolic problems Chapter 5 Renormalized Solutions 5.1 Problems With Irregular Data 5.1.1 Consequences of mere integrability of data 5.1.2 Various notions of solutions 5.1.3 Comments on the scheme of the proof of existence 5.2 Renormalized Solutions to Elliptic Problems 5.2.1 Formulation of the problem 5.2.2 Existence and uniqueness 5.2.3 Exercises 5.3 Renormalized Solutions to Parabolic Problems 5.3.1 Formulation of the problem 5.3.2 Approximation in time 5.3.3 The comparison principle 5.3.4 Existence and uniqueness 5.3.5 Exercises Chapter 6 Homogenization of Elliptic Boundary Value Problems 6.1 Formulation of the Homogenization Problem 6.2 Definitions, Main Result and the Strategy 6.3 The Functional Setting 6.4 Homogenization Tools in the Setting of Musielak–Orlicz Spaces 6.5 Properties of the Cell Problem 6.6 The Homogenized Operator and the Limit Problem 6.7 Existence of Solutions for a Fixed ε 6.8 Limit Passage to the Homogenized Problem Chapter 7 Non-Newtonian Fluids 7.1 Introducing the Problem 7.2 Heat-Conducting Non-Newtonian Fluids 7.2.1 A few words about notation 7.2.2 Existence of weak solutions. Formulation of the problem 7.2.3 The proof of existence of weak solutions 7.3 A Generalized Stokes System 7.3.1 Formulation of the problem and the existence result 7.3.2 Domains and closures 7.3.3 The proof of existence 7.4 Local Pressure and the Fluid-Structure Interaction Problem for Non-Newtonian Fluids 7.4.1 Decomposition of the pressure function and local estimates 7.4.2 Motion of rigid bodies in non-Newtonian fluid. An application of the method Part III Auxiliaries Chapter 8 Basics 8.1 Measure Theory 8.2 Functional Analysis 8.3 Approximation Chapter 9 Functional Inequalities 9.1 Sobolev-Type Embedding 9.2 The Korn Inequality References List of Symbols Index